Answer to Question #179834 in Calculus for Johannes Steven

Question

1. Consider the relation P = {(1,7), (-1,7), (3,9), (1,3). is P a function? Justify your answer.
2. Use sign table to determine the values of x for which x²+ 9x +20 is less than or equals to 0
3. Prove that f(x) = x² + 2x is not injective

Answer 1

Q1.

A relation from set A to set B will be a mapping if every element of set A is uniquely related to the elements of B. But in the P  = {(1,7), (-1,7), (3,9), (1,3)} , 1 is related to 7 and 3. So P is not a mapping.

Q2.

x²+9x+20

= x²+5x+4x+20

=x(x+5)+4(x+5)

=(x+5)(x+4)

So x+5 = 0 => x = -5

x+4 = 0 => x =-4

Now from the sign table below

x²+9x+20 is positive if x < -5 or x > -4,

x²+9x+20 is negative if -5<x<-4

and x²+9x+20 is zero if x = -5, -4

So

x²+9x+20 is less than or equal to zero if -5 ≤ x ≤ -4 i.e. if x $\isin[-5,-4]$

Q3.

f(x)=x²+2x

f(x) = f(y) for x, y in the domain of f

=> x²+2x = y²+2y

=> x²-y² + 2(x-y) = 0

=> (x-y)(x+y+2)=0

So x=y and x= -y-2 are two solutions.

For example f(0) = 0 and f(-2)= 4-4=0

So 0 and -2 have same image 0.

Therefore f(x) = x²+2x is not injective.